Episode Eleven – Newton’s Achievements

Binomial Theorem

This mathematical theorem involves the expansion of powers of binomials. A binomial is something like “X squared plus 2”. Basically, the binomial theorem deals with some variable plus something else, all to the power of something.

These powers of binomials can be written in their full form as a polynomial. Such as x squared plus 5x + 3.

Let us suppose you have something like “2x +3”. And you want to multiply it by itself, you have this:

(2x+3)2

You get a polynomial like this:

4x2 + 12x + 9

The binomial theorem lets you find these polynomials.

So, if you want to multiply (2x + 3) by itself 5 times, what do you get? Well, the binomial theorem will tell you that.

The Binomial Theorem uses Pascal’s Triangle, which was known to ancient mathematicians.

Which is good, because the polynomial will look something like 32x5 + Nx4….etc. Calculating this by hand would take a while.

Now, it should be noted that others, including Euclid and Al-Karaji had developed less generalized methods for doing this kind of thing. However, Newton’s method was more general. It could also be applied to all real numbers, as opposed to only nonnegative integers.

But, so what? Of what importance or application is this to the real world? Well, quite a lot as it turns out.

Every computer on most networks, including the internet, has an IP address. This is essentially a unique number that is used to identify your particular computer on the network (I am over-simplifying a bit but the fine details do not matter that much).

Well, these are often automatically assigned and the binomial theorem helps to do that. The binomial theorem can deal with other such network problems.

It is also used in estimating probabilities in fields like economics and weather forecasts.. It is used to help design infrastructure to help find the proper amount of materials to use. As well as that, it also helps rank things. And so on….